I have a 4 node SQL 2012 environment with over 2,000 DBs and all SQL instance AO/AGs replicate to one of the nodes which is marked as 2ndary for all. All backups are taken on this 2ndary server. We generally only failover to the 2ndary for maintenance/updates on the primarys or if there is an outage on a primary. I took over this environment a few months ago and I have to say I haven't tested whether I can perform backups while an AG is failed over to the 2ndary. Are your AGs set to Synchronous Commit - Availability Mode and are they set as Readable Secondary?
I realize the original post is a year old and perhaps you opened up a support incident. I thought I'd comment anyway.
